Is it possible to use vertex paint with symmetry? Mirror modifier not an option
Spoiler alert, this is only a workaround, there's no native way to paint with symmetry on vertex paint mode, :'(

1) Make a vertex group assigned to one half of the symmetrical model

2) Duplicate-link(pressing "alt-D") the object and mirror the duplicate with "Ctrl + m" and then press "x"

3) Add a "Data Transfer" modifier to the original object

4) Set the source object as the duplicated object

5) Tick "Face Corner Data" and select Vcol, tick on Generate Data Layer at the bottom of the modifier

6) Set the vertex group to the previous vertex group created in 1)

7) You must see the original model with the vertex colors symmetrical and live painting with symmetry, you could apply the modifier when you have finished painting.

8) If the results are not as expected you can play with the method of mapping the source, the default method is "Nearest Corner And Best Matching Normal"

Modifier not viewport seen


Modifier viewport seen, live painting symmetrical vertex colors


PD = I have heard that is pretty hard to implement a x-mirror option for vertex paint mode because vertex color is face-corner data, but I don't believe that, :(